Statistical Analysis of Crossed Undulator for Polarization Control in a SASE FEL

0.19 MB·English

by  Yuantao Ding

Statistical Analysis of Crossed Undulator for Polarization Control
in a SASE FEL
Yuantao Ding and Zhirong Huang
Stanford Linear Accelerator Center, Menlo Park, CA 94025
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0 Abstract
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2 There is a growing interest in producing intense, coherent x-ray radiation with an adjustable

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a and arbitrary polarization state. In this paper, we study the crossed undulator scheme (K.-J.
J
0 Kim, Nucl. Instrum. Methods A 445, 329 (2000)) for rapid polarization control in a self-amplified
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spontaneous emission (SASE) free electron laser (FEL). Because a SASE source is a temporally

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chaotic light, we perform a statistical analysis on the state of polarization using FEL theory and
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c simulations. We show that by adding a small phase shifter and a short (about 1.3 times the FEL
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power gain length), 90 rotated planar undulator after the main SASE planar undulator, one can
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obtain circularly polarized light – with over 80% polarization – near the FEL saturation.
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PACS numbers: 41.60.Cr
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I. INTRODUCTION
Several x-ray free electron lasers (FELs) based on self-amplified spontaneous emission
(SASE) are being developed worldwide as next-generation light sources [1, 2, 3]. In the soft
x-ray wavelength region, polarization control (from linear to circular) is highly desirable in
studying ultrafast magentic phenomena and material science. The x-ray FEL is normally
linearly polarized based on planar undulators. Variable polarization could in principle be
provided by employing an APPLE-type undulator [4]. However, its mechanical tolerance
for lasing at x-ray wavelengths has not been demonstrated, and its focusing property may
changesignificantly whenitspolarizationisaltered. Analternativeapproachforpolarization
control is the so-called “crossed undulator” (or “crossed-planar undulator”), which is the
subject of this paper.
The crossed-planar undulator was proposed by K.-J Kim to generate arbitrarily polar-
ized light in synchrotron radiation [5] and FEL sources [6]. It is based on the interference
of horizontal and vertical radiation fields generated by two adjacent planar undulators in a
crossed configuration (see Fig. 1). A phase shifter between the undulators is used to delay
the electron beam and hence to control the final polarization state. For incoherent radiation
sources, the radiation pulses generated in two adjacent undulators by each electron do not
overlap in time. Thus, a monochromator after the second undulator is required to stretch
both pulses temporally in order to achieve interference. The degree of polarization is limited
by beam emittance, energy spread, and the finite resolution of the monochromator, as stud-
ied in a series of experiments at BESSY [7, 8]. On the other hand, for completely coherent
radiation sources (such as generated from a seeded FEL amplifier or an FEL oscillator),
the interference occurs due to the overlap of two radiation components in the second un-
dulator [6]. A recent crossed-undulator experiment at the Duke storage ring FEL reported
controllable polarization switches with a nearly 100% total degree of polarization [9].
It is well-known in the FEL community that SASE light is transversely coherent but
temporally chaotic due to the shot noise startup. Thus, the effectiveness of the crossed
undulator for polarization control deserves a detailed study. In this paper, starting with
one-dimensional (1D) FEL theory, we calculate both radiation components and generalize
the results of Ref. [6] to the case of SASE. We then determine the required length of the
second undulator in order to produce the same average power as that produced in the
2
y
x
E
y
z
E
x
Phase
Undulator-1 Undulator-2
Shifter
FIG. 1: (color) Schematic of the crossed undulator for polarization control
first undulator. We show that the degree of polarization can be determined by the time
correlation of the two radiation fields and compute its asymptotic expression in the high-
gain limit. The analytical results are compared with 1D SASE simulations after a proper
statistical averaging. Finally, three-dimensional (3D) effects and simulation results are also
discussed.
II. FIELD CALCULATION
Figure 1 shows a schematic of the crossed undulator applied to a SASE FEL. In the
first planar undulator with a total length L , spontaneous radiation is amplified to generate
1
horizontally polarized SASE field E . In the second undulator (of length L ) that is rotated
x 2
90 with respect to the first one, E propagates freely without interacting with the electron
◦ x
beam, while avertically polarizedradiationfieldE isproducedby themicro-bunched beam.
y
A simple phase shifter such as a four-dipole chicane placing between the two undulators can
slightly delay the electrons in order to adjust the relative phase of the two polarization
components.
In this section, we determine both SASE field components generated by the crossed
undulator. Let E(z,t) be the complex but slowly varying electric field at undulator distance
z and time t. We write
ω dν
E(z,t) = 1 E (z)ei∆ν[(k1+ku)z ω1t], (1)
ν −
√2π
Z
where ω = k c is the fundamental resonant frequency corresponding to the average beam
1 1
energy (c is the speed of light); ν = ω/ω and ∆ν = ν 1 is the relative frequency detuning,
1
−
k = 2π/λ with λ the undulator period. Following Refs. [10, 11], the 1D FEL interaction
u u u
starting from shot noise can be described by the coupled Maxwell-Klimontovich equations.
3
In the small signal regime before FEL saturation, the equations can be linearized and solved
by the Laplace transformation:
dµ
E (z) = ( i2ρk )e i2ρµkuzE ,
ν u − ν,µ
2πi −
I
dµ κ E dV/dη F (0)
F (z) = e i2ρµkuz 1 ν,µ − ν , (2)
ν −
2πi (η/ρ µ)
I −
where
i iκ n F (0)
2 0 ν
E = E (0)+ dη ,
ν,µ ν
2ρk D(µ) 2ρk η/ρ µ
u (cid:18) u Z − (cid:19)
∆ν V(η)
D(µ) = µ dη . (3)
− 2ρ − (η/ρ µ)2
Z −
Here E and F are, respectively, the Fourier components of the electric field and of the
ν ν
Klimontovich distributionfunctionthatdescribes thediscrete electrons inlongitudinalphase
space, with E (0) and F (0) the Fourier components of the initial conditions; D(µ) = 0
ν ν
determines the FEL dispersion relation where µ is the Laplace parameter. In addition,
parameter ρ is the dimensionless FEL Pierce parameter [12], V(η) is the electron energy
distribution with η the relative energy deviation, n is the electron volume density; κ =
0 1
eK[JJ]/(4γ2mc2), κ = eK[JJ]/(2ǫ γ ), where K is the dimensionless undulator strength
0 2 0 0
parameter, the Bessel function factor [JJ] is equal to [J (ξ) J (ξ)] with ξ = K2/(4+2K2),
0 1
−
γ is the initial electron energy in units of mc2, and ǫ is the vacuum permittivity. Note
0 0
that the contour integration of µ in Eq. (2) must enclose all singularities in the complex µ
plane. Based on this solution, we can calculate radiation field components in the crossed
undulator according to their initial conditions.
A. Horizontal radiation field
The radiation field E in the first undulator develops from electron shot noise, with the
x
initial conditions
1 Ne
Ex(0) = 0, Fx(0)dη = eiνω1tj(0), (4)
ν ν N
λ
Z j=1
X
where N is the number of electrons in one radiation wavelength, and t (0) is the random
λ j
arrival time of the jth electron at the entrance to the first undulator. We assume the first
4
undulator operates in the exponential growth regime. In this regime, the dispersion relation
has a solution µ with a positive imaginary part that gives rise to an exponentially growing
0
field amplitude. For a cold beam with vanishing energy spread, we take V(η) = δ(η) in
Eq. (2) and obtain
iκ n Ne
Ex(z) = − 2 0 e iµ02ρkuz eiνω1tj for z L . (5)
ν 2ρk N 3µ − ≤ 1
u λ 0
j=1
X
In this high-gain regime, the electron distribution from Eq. (2) can be simplified as [6]:
iκ Ex(z)dV/dη
Fx(z) = 1 ν for z L . (6)
ν 2k ( µ ρ+η) ≤ 1
u 0
−
This electron distribution function will be used as an initial condition for the calculation of
the vertical radiation field as follows.
B. Vertical radiation field
The radiation field E in the second undulator is generated by the pre-bunched electron
y
beam in the first undulator. To control the radiation polarization, the required path length
delay of the phase shifter chicane is on the order of the FEL wavelength. Such a weak
chicane does not have dispersive effects that could result in micro-bunching, such as can be
found for example in an optical klystron (see, e.g., Ref. [13]). Hence, the initial conditions
at the entrance of the second undulator is
Ey(0) = 0, Fy(0) = Fx(L ). (7)
ν ν ν 1
As the electron beam develops micro-bunching during the FEL interaction in the first
undulator, it will radiate coherently in the second undulator. From discussions in Ref. [6]
and simulation results shown in Sec. IV below, the intensity of E can increase to the same
y
level as that of E in about one gain length. Thus, for a relatively short second undulator,
x
we consider only coherent radiation and ignore any feedback of the radiation on the electron
beam. With this approximation, the third term at the right hand side of D(µ) in Eq. (3)
can be dropped, and Eq. (2) can now be written as
dµ e i2ρµkuz2 iκ n Fx(L )
Ey(z ) =eiφ − 2 0 dη ν 1
ν 2 2πiµ ∆ν/2ρ 2ρk η/ρ µ
I − (cid:20) u Z − (cid:21)
dµ e i2ρµkuz2 Ex(L )µ+µ
= eiφ − ν 1 0 . (8)
− 2πiµ ∆ν/2ρ µ2 µ2
I − 0
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Here z is the undulator distance from the beginning of the second undulator. The extra
2
phase factor eiφ is introduced by the phase shifter just before the second undulator. In the
last step of Eq. (8), we have taken a cold beam with vanishing energy spread and made
use of the relation κ κ n = 4k2ρ3. Note that µ is the exponential growth solution that
1 2 0 u 0
satisfies D(µ ) = 0 and is a function of the detuning parameter ∆ν, i.e.,
0
1 ∆ν (∆ν)2 √3 (∆ν)2
µ 1 + +i 1 . (9)
0 ≈ −2 − 3ρ 36ρ2 2 − 36ρ2
(cid:20) (cid:21) (cid:20) (cid:21)
Eq. (8) can be solved by the residue theorem:
ψ 2i
Ey(z ) = Ex(L )ei(φ ψ/2)sinc ρk z µ eiα(ρk z )2 , (10)
ν 2 ν 1 − 2 µ2 u 2 − 0 u 2
(cid:18) (cid:19) 0
(cid:2) (cid:3)
where sinc(x) = sin(x)/x, ψ = ∆νk z , and
u 2
sin(ψ/2)
α = arctan . (11)
sinc(ψ/2) cos(ψ/2)
(cid:20) − (cid:21)
Note that α = π/2 when ∆ν = 0. The first term in the square bracket of Eq. (10) describes
coherent spontaneous radiation from a density-modulated beam and grows linearly with
the undulator distance z (as discussed in Ref. [14] in the context of harmonic generation).
2
Since the electron beam from the first undulator possesses not only density modulation but
also energy modulation, the momentum compaction of the second undulator can convert
the energy modulation into additional density modulation. Thus, the second term in the
square bracket of Eq. (10) describes the enhanced radiation due to the evolution of the den-
sity modulations inside the second undulator which grows quadratically with the undulator
distance.
In order to generate circularly polarized light, we require that both E and E have the
x y
same average amplitude. From Eq. (10), this corresponds to the condition
2i
ρk z µ eiα(ρk z )2 = 1. (12)
µ2 u 2 − 0 u 2
(cid:12) 0 (cid:12)
(cid:12) (cid:2) (cid:3)(cid:12)
We consider a cold electron(cid:12)beam with vanishing ener(cid:12)gy spread, hence the growth rate
(cid:12) (cid:12)
Im(µ ) is maximized on resonance, i.e., ∆ν = 0. In this case we obtain the required length
0
of the second undulator from Eq.(12)
λ
u
L 1.3L , where L = (13)
2 G G
≈ 4π√3ρ
is the 1D power gain length.
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III. DEGREE OF POLARIZATION
The interference of the two radiation components generated by the crossed undulator will
produce flexible polarization. At the end of the second undulator when z = L +L , these
1 2
radiation fields in the time domain are
ω dν
E (t) = 1 Ey(z = L )ei∆ν[(k1+ku)(L1+L2) ω1t],
y √2π ν 2 2 −
Z
ω dν
E (t) = 1 Ex(z = L )ei∆ν[(k1+ku)L1+k1L2 ω1t]. (14)
x √2π ν 1 −
Z
Note that we only used Eq. (1) for E at z = L (and t ) and applied the free space
x 1 1
propagation phase factor ei∆ν[k1L2 ω1(t t1)] in the second undulator as E does not interact
− − x
with the electron beam there. Because of the chaotic nature of SASE radiation, we perform
a statistical analysis to quantify the state of polarization.
Following the standard optics textbooks (see, e.g., Refs. [15, 16]), the state of polarization
can be described by the coherency matrix
E (t)E (t) E (t)E (t)
J = h x x∗ i h x y∗ i , (15)
 
E (t)E (t) E (t)E (t)
h y x∗ i h y y∗ i
 
where * means complex conjugate, and the angular bracket refers to the ensemble average.
The degree of polarization can be calculated as [15, 16]
det[J]
P 1 4 , (16)
≡ s − (tr[J])2
where det[J] and tr[J] are the determinant and trace of the coherency matrix, respectively.
It is also convenient to introduce the first-order time correlation between E and E as
x y
E (t)E (t)
g h x y∗ i . (17)
xy ≡ [ E (t) 2 E (t) 2 ]1/2
x y
h| | ih| | i
For polarization control in the crossed undulator, we are particularly interested in the
case when the average intensities of the two radiation components arethe same: E (t) 2 =
x
h| | i
E (t) 2 = I¯. Under this condition, the coherency matrix simplifies to
y
h| | i
1 g eiθ
J = I¯ | xy| , (18)
 g e iθ 1 
xy −
| |
 
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where θ is the phase difference between E and E . When θ = π, the combined radiation
x y ±2
is circularly polarized; when θ = 0 or π, it is linearly polarized at 45 relative to the
◦
±
horizontal axis. The state of polarization is controllable by adjusting the phase shift φ in
Eq. (10) so that the net phase in g is θ = π or 0/π. With equal intensity in both
xy ±2
transverse directions, the degree of polarization in Eq. (16) is simply given by the amplitude
of the x-y time correlation, i.e.,
P = g . (19)
xy
| |
In the x-ray wavelength region, the electron bunch duration is typically much longer than
the coherence time of the SASE radiation. Thus, a SASE pulse consists of many random
intensity spikes that are statistically independent. For a flattop current distribution (of
width T), we can convert the ensemble average of Eq. (17) into a time average as
1 T/2
g = lim dtE (t)E (t)
xy T I¯T x y∗
→∞ Z T/2
−
1
= ∞ ω dνEx(L )Ey (L )e i∆νkuL2, (20)
I¯T 1 ν 1 ν∗ 2 −
Z
−∞
where we have applied Eq. (14) and the Parseval relation in converting the time integration
to the frequency integration. Assuming that the first undulator operates in the exponential
gain regime, the frequency dependence of Ex is approximately Gaussian, i.e.,
ν
I¯T (∆ν)2
Ex(z) 2 = e− 2σν2 , (21)
h| ν | i √2πσ
ω
where the relative rms SASE bandwidth is [10, 11]
9ρ
σ = σ /ω = . (22)
ν ω 1
s√3kuL1
Since the short second undulator generates coherent radiation from a pre-bunched beam
that possesses the same narrow bandwidth σ , we can expand µ2 in Eq. (10) to first order
ν 0
in ∆ν by using Eq. (9). We also ignore the frequency dependence of the second term in the
square bracket of Eq. (10) because its contribution to the radiation intensity is relatively
small. Finally, we have
exp ν¯2 iν¯σνkuL2 sinc ν¯σνkuL2
1 ∞ − 2 − 2 2
g dν¯ , (23)
| xy| ≈ √2π(cid:12) (cid:16) 1+( 1 +i(cid:17)√3)ν¯σν(cid:0) (cid:1)(cid:12)
(cid:12)Z−∞ −2 2 3ρ (cid:12)
(cid:12) (cid:12)
where ν¯ = ∆ν/σ . In view (cid:12)of Eq. (13), we take L = 1.3L in Eq. (2(cid:12)3) and obtain the
ν (cid:12) 2 G (cid:12)
degree of polarization by computing g .
xy
| |
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TABLE I: Main parameters for the LCLS soft x-ray FEL used in simulations.
Parameter value unit
electron beam energy 4.3 GeV
relative energy spread 0(0.023) %
bunch peak current 2 kA
transverse norm. emittance 1.2 µm
average beta function 8 m
undulator period λ 3 cm
u
undulator parameter K 3.5
FEL wavelength 1.509 nm
FEL ρ parameter 0.119 %
1D power gain length L 1.17 m
G
3D power gain length L3D 1.48 m
G
IV. NUMERICAL SIMULATIONS
A. 1D results
We first use a 1D FEL code to simulate the SASE radiation produced by the crossed
undulator configuration and to analyze the degree of polarization. The code follows the
time-dependent approach developed in Ref. [17] and employs the shot noise algorithm of
Penman and McNeil [18]. Electron energy spread can be included using Fawley’s beamlet
method [19]. After computing the E field produced in the first undulator, we allow E to
x x
propagate freely without further interacting with the electron beam. The simulated electron
distribution from the first undulator is then used to generate the E field in the second
y
undulator.
As a numerical example, we use the parameter set listed in Table I that is similar to the
soft x-ray LCLS operation [1]. In the 1D simulations, the energy spread is set to zero since
we want to compare with the previous analytical results.
Fig. 2 shows the average radiation power in both x and y directions produced by the
cross undulator. The length of the first undulator is allowed to vary, while the second
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2
10
)
W 100
G
(
r
e
w −2
10
o
p
e
g
a
ver10−4 <P >
A
x
<P >
y
−6
10
0 5 10 15 20 25
L ( m )
1
FIG. 2: (color) 1D simulations of theaverage SASE power at 1.5 nm from the first(bluecross) and
the second (red plus) undulator. Here L is the length of the first undulator, L = 1.3L = 1.53
1 2 G
m is the length of the second undulator.
undulator length L = 1.3L 1.53 m is held constant. As predicted by Eq. (13), the
2 G
≈
power of the two radiation components are essentially the same in the exponential gain
regime. Near saturation, the power of the vertical field is lower than that of the horizontal
one because the FEL-induced energy spread starts to de-bunch the electron beam in the
second undulator. We repeat the simulations 200 times for each L with different random
1
seeds to start the process and calculate the first-order time correlation between E and E
x y
at the exit of the second undulator using the ensemble average defined in Eq. (17). Figure 3
shows the amplitude of this correlation from the simulation results as well as the numerical
integration of Eq. (23) (the red solid curve) for a comparison. When the first undulator
is less than a couple of gain lengths, the crossed undulator operates in the spontaneous
emission regime, the amplitude of the x-y correlation and hence the degree of polarization
are very small without the use of a monochromator. The degree of polarization increases in
the exponential growth regime and reaches a maximum of 85% near the FEL saturation. In
this regime and especially when the gain is very high, we see very good agreement between
simulations and Eq. (23). In the saturation regime, the amplitude of the x-y correlation
starts to decrease, and the linear theory starts to deviate from the simulation results.
There are two effects that prevent the degree of polarization to reach 100% in a crossed-
undulator SASE FEL. First, there is relative slippage between E and E in the second
x y
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